f(x)=x/g(x). If f and g are differentiable, f'(x)=(g(x)-xg'(x))/(g(x))². But if g(0)=0, then f'(0) cannot be defined, so f(x) is not differentiable at x=0, even if f(0)=0. Alternatively, g(x) is not differentiable at x=0, even if g(0)=0, because g'(0) cannot be defined. So f(0)=g(0)=0 implies that f and g are un-differentiable at x=0.
The statement is false.